| We consider a ring cavity which is filled with absorbing two-level atoms and driven by an external coherent field tuned at near-resonance with both the atomic transition frequency and one of the cavity resonances. We derive from first principles a system of ordinary differential equations describing the evolution of the cavity field and the atomic medium which is assumed in general to be inhomogeneously broadened. We then focus on the special case of a homogeneously broadened medium to investigate the possibility of coexistence of two components of the cavity field propagating in opposite directions, the one co-propagating with the external field being called the forward field, and the other the backward field. We find that, apart from the well-known unidirectional steady states where only the forward field exists, novel steady states appear, for properly selected parameters, where both the forward and backward fields exist but oscillate at different frequencies. These steady states are thus called nonsynchronous steady states. Our linear stability analysis shows that the existence of a nonsynchronous steady state is often associated with instabilities of the backward field of a unidirectional steady state although that is not always the case due to hysteresis. Of special interest are the cases where instabilities of the backward field but not of the forward field occur. To unravel the physical origin of the instabilities of the backward field of a unidirectional steady state, we implement the weak sideband approach which establishes a clear connection between these instabilities and the Mollow spectrum and also help us understand the frequency difference between the forward and bacward fields in a nonsynchronous steady state. The application of this approach suggests the existence of a seemingly universal connection between the gain mechanism responsible for an instability of the backward field of a unidirectional steady state and the way the forward field behaves in response to the growth of the backward field due to the instability. Our numerical simulation of the time-dependent solutions to the equations of motion depicts the emergence of nonsynchronous steady states in time and also verifies their stability. |
